LMFDB - Embedded newform 1656.2.a.j.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1656,2,Mod(1,1656)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1656, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1656.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1656 = 2^{3} \cdot 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1656.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$13.2232265747$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 184)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	1656.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{5} +O(q^{10})$	$q-2.00000 q^{5} +3.12311 q^{11} +0.438447 q^{13} -5.12311 q^{17} -3.12311 q^{19} +1.00000 q^{23} -1.00000 q^{25} -3.56155 q^{29} -2.43845 q^{31} +8.24621 q^{37} +9.80776 q^{41} -8.00000 q^{43} +0.684658 q^{47} -7.00000 q^{49} -2.00000 q^{53} -6.24621 q^{55} -10.2462 q^{59} -4.24621 q^{61} -0.876894 q^{65} +3.12311 q^{67} -13.5616 q^{71} -14.6847 q^{73} +3.12311 q^{79} -14.2462 q^{83} +10.2462 q^{85} -11.3693 q^{89} +6.24621 q^{95} +11.3693 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{5}+O(q^{10}) $ 2 * q - 4 * q^5	$ 2 q - 4 q^{5} - 2 q^{11} + 5 q^{13} - 2 q^{17} + 2 q^{19} + 2 q^{23} - 2 q^{25} - 3 q^{29} - 9 q^{31} - q^{41} - 16 q^{43} - 11 q^{47} - 14 q^{49} - 4 q^{53} + 4 q^{55} - 4 q^{59} + 8 q^{61} - 10 q^{65} - 2 q^{67} - 23 q^{71} - 17 q^{73} - 2 q^{79} - 12 q^{83} + 4 q^{85} + 2 q^{89} - 4 q^{95} - 2 q^{97}+O(q^{100}) $ 2 * q - 4 * q^5 - 2 * q^11 + 5 * q^13 - 2 * q^17 + 2 * q^19 + 2 * q^23 - 2 * q^25 - 3 * q^29 - 9 * q^31 - q^41 - 16 * q^43 - 11 * q^47 - 14 * q^49 - 4 * q^53 + 4 * q^55 - 4 * q^59 + 8 * q^61 - 10 * q^65 - 2 * q^67 - 23 * q^71 - 17 * q^73 - 2 * q^79 - 12 * q^83 + 4 * q^85 + 2 * q^89 - 4 * q^95 - 2 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.12311	0.941652	0.470826	−	0.882226i	$-0.343956\pi$
$11$	3.12311	0.941652	0.470826	+	0.882226i	$0.343956\pi$
$12$	0	0
$13$	0.438447	0.121603	0.0608017	−	0.998150i	$-0.480634\pi$
$13$	0.438447	0.121603	0.0608017	+	0.998150i	$0.480634\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.12311	−1.24254	−0.621268	−	0.783598i	$-0.713382\pi$
$17$	−5.12311	−1.24254	−0.621268	+	0.783598i	$0.713382\pi$
$18$	0	0
$19$	−3.12311	−0.716490	−0.358245	−	0.933628i	$-0.616625\pi$
$19$	−3.12311	−0.716490	−0.358245	+	0.933628i	$0.616625\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−3.56155	−0.661364	−0.330682	−	0.943742i	$-0.607279\pi$
$29$	−3.56155	−0.661364	−0.330682	+	0.943742i	$0.607279\pi$
$30$	0	0
$31$	−2.43845	−0.437958	−0.218979	−	0.975730i	$-0.570273\pi$
$31$	−2.43845	−0.437958	−0.218979	+	0.975730i	$0.570273\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.24621	1.35567	0.677834	−	0.735215i	$-0.262919\pi$
$37$	8.24621	1.35567	0.677834	+	0.735215i	$0.262919\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	9.80776	1.53172	0.765858	−	0.643010i	$-0.222315\pi$
$41$	9.80776	1.53172	0.765858	+	0.643010i	$0.222315\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.684658	0.0998677	0.0499338	−	0.998753i	$-0.484099\pi$
$47$	0.684658	0.0998677	0.0499338	+	0.998753i	$0.484099\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	−6.24621	−0.842239
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−10.2462	−1.33394	−0.666972	−	0.745083i	$-0.732410\pi$
$59$	−10.2462	−1.33394	−0.666972	+	0.745083i	$0.732410\pi$
$60$	0	0
$61$	−4.24621	−0.543672	−0.271836	−	0.962344i	$-0.587631\pi$
$61$	−4.24621	−0.543672	−0.271836	+	0.962344i	$0.587631\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−0.876894	−0.108765
$66$	0	0
$67$	3.12311	0.381548	0.190774	−	0.981634i	$-0.438900\pi$
$67$	3.12311	0.381548	0.190774	+	0.981634i	$0.438900\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−13.5616	−1.60946	−0.804730	−	0.593641i	$-0.797690\pi$
$71$	−13.5616	−1.60946	−0.804730	+	0.593641i	$0.797690\pi$
$72$	0	0
$73$	−14.6847	−1.71871	−0.859355	−	0.511380i	$-0.829134\pi$
$73$	−14.6847	−1.71871	−0.859355	+	0.511380i	$0.829134\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	3.12311	0.351377	0.175688	−	0.984446i	$-0.443785\pi$
$79$	3.12311	0.351377	0.175688	+	0.984446i	$0.443785\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−14.2462	−1.56372	−0.781862	−	0.623451i	$-0.785730\pi$
$83$	−14.2462	−1.56372	−0.781862	+	0.623451i	$0.785730\pi$
$84$	0	0
$85$	10.2462	1.11136
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−11.3693	−1.20515	−0.602573	−	0.798064i	$-0.705858\pi$
$89$	−11.3693	−1.20515	−0.602573	+	0.798064i	$0.705858\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	6.24621	0.640848
$96$	0	0
$97$	11.3693	1.15438	0.577190	−	0.816610i	$-0.304149\pi$
$97$	11.3693	1.15438	0.577190	+	0.816610i	$0.304149\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−12.2462	−1.21854	−0.609272	−	0.792961i	$-0.708538\pi$
$101$	−12.2462	−1.21854	−0.609272	+	0.792961i	$0.708538\pi$
$102$	0	0
$103$	−14.2462	−1.40372	−0.701860	−	0.712314i	$-0.747647\pi$
$103$	−14.2462	−1.40372	−0.701860	+	0.712314i	$0.747647\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−11.1231	−1.07531	−0.537656	−	0.843165i	$-0.680690\pi$
$107$	−11.1231	−1.07531	−0.537656	+	0.843165i	$0.680690\pi$
$108$	0	0
$109$	13.1231	1.25697	0.628483	−	0.777824i	$-0.283676\pi$
$109$	13.1231	1.25697	0.628483	+	0.777824i	$0.283676\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2.87689	0.270635	0.135318	−	0.990802i	$-0.456794\pi$
$113$	2.87689	0.270635	0.135318	+	0.990802i	$0.456794\pi$
$114$	0	0
$115$	−2.00000	−0.186501
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1.24621	−0.113292
$122$	0	0
$123$	0	0
$124$	0	0
$125$	12.0000	1.07331
$126$	0	0
$127$	0.684658	0.0607536	0.0303768	−	0.999539i	$-0.490329\pi$
$127$	0.684658	0.0607536	0.0303768	+	0.999539i	$0.490329\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−3.31534	−0.289663	−0.144831	−	0.989456i	$-0.546264\pi$
$131$	−3.31534	−0.289663	−0.144831	+	0.989456i	$0.546264\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.87689	−0.587533	−0.293766	−	0.955877i	$-0.594909\pi$
$137$	−6.87689	−0.587533	−0.293766	+	0.955877i	$0.594909\pi$
$138$	0	0
$139$	−3.31534	−0.281204	−0.140602	−	0.990066i	$-0.544904\pi$
$139$	−3.31534	−0.281204	−0.140602	+	0.990066i	$0.544904\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.36932	0.114508
$144$	0	0
$145$	7.12311	0.591542
$146$	0	0
$147$	0	0
$148$	0	0
$149$	20.2462	1.65863	0.829317	−	0.558778i	$-0.188730\pi$
$149$	20.2462	1.65863	0.829317	+	0.558778i	$0.188730\pi$
$150$	0	0
$151$	−10.4384	−0.849469	−0.424734	−	0.905318i	$-0.639633\pi$
$151$	−10.4384	−0.849469	−0.424734	+	0.905318i	$0.639633\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4.87689	0.391722
$156$	0	0
$157$	21.1231	1.68581	0.842904	−	0.538064i	$-0.180844\pi$
$157$	21.1231	1.68581	0.842904	+	0.538064i	$0.180844\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	18.9309	1.48278	0.741390	−	0.671074i	$-0.234167\pi$
$163$	18.9309	1.48278	0.741390	+	0.671074i	$0.234167\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	0	0
$169$	−12.8078	−0.985213
$170$	0	0
$171$	0	0
$172$	0	0
$173$	10.0000	0.760286	0.380143	−	0.924928i	$-0.375875\pi$
$173$	10.0000	0.760286	0.380143	+	0.924928i	$0.375875\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	3.31534	0.247800	0.123900	−	0.992295i	$-0.460460\pi$
$179$	3.31534	0.247800	0.123900	+	0.992295i	$0.460460\pi$
$180$	0	0
$181$	3.75379	0.279017	0.139508	−	0.990221i	$-0.455448\pi$
$181$	3.75379	0.279017	0.139508	+	0.990221i	$0.455448\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−16.4924	−1.21255
$186$	0	0
$187$	−16.0000	−1.17004
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−25.3693	−1.83566	−0.917830	−	0.396974i	$-0.870060\pi$
$191$	−25.3693	−1.83566	−0.917830	+	0.396974i	$0.870060\pi$
$192$	0	0
$193$	−0.438447	−0.0315601	−0.0157801	−	0.999875i	$-0.505023\pi$
$193$	−0.438447	−0.0315601	−0.0157801	+	0.999875i	$0.505023\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	16.9309	1.20627	0.603137	−	0.797637i	$-0.293917\pi$
$197$	16.9309	1.20627	0.603137	+	0.797637i	$0.293917\pi$
$198$	0	0
$199$	11.1231	0.788496	0.394248	−	0.919004i	$-0.371005\pi$
$199$	11.1231	0.788496	0.394248	+	0.919004i	$0.371005\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−19.6155	−1.37001
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−9.75379	−0.674684
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	16.0000	1.09119
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−2.24621	−0.151097
$222$	0	0
$223$	−28.4924	−1.90799	−0.953997	−	0.299817i	$-0.903075\pi$
$223$	−28.4924	−1.90799	−0.953997	+	0.299817i	$0.903075\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	6.24621	0.414576	0.207288	−	0.978280i	$-0.433536\pi$
$227$	6.24621	0.414576	0.207288	+	0.978280i	$0.433536\pi$
$228$	0	0
$229$	14.8769	0.983093	0.491546	−	0.870851i	$-0.336432\pi$
$229$	14.8769	0.983093	0.491546	+	0.870851i	$0.336432\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	16.0540	1.05173	0.525865	−	0.850568i	$-0.323742\pi$
$233$	16.0540	1.05173	0.525865	+	0.850568i	$0.323742\pi$
$234$	0	0
$235$	−1.36932	−0.0893244
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−26.0540	−1.68529	−0.842646	−	0.538468i	$-0.819003\pi$
$239$	−26.0540	−1.68529	−0.842646	+	0.538468i	$0.819003\pi$
$240$	0	0
$241$	6.49242	0.418214	0.209107	−	0.977893i	$-0.432944\pi$
$241$	6.49242	0.418214	0.209107	+	0.977893i	$0.432944\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	14.0000	0.894427
$246$	0	0
$247$	−1.36932	−0.0871275
$248$	0	0
$249$	0	0
$250$	0	0
$251$	6.24621	0.394257	0.197129	−	0.980378i	$-0.436838\pi$
$251$	6.24621	0.394257	0.197129	+	0.980378i	$0.436838\pi$
$252$	0	0
$253$	3.12311	0.196348
$254$	0	0
$255$	0	0
$256$	0	0
$257$	5.31534	0.331562	0.165781	−	0.986163i	$-0.446986\pi$
$257$	5.31534	0.331562	0.165781	+	0.986163i	$0.446986\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	19.1231	1.17918	0.589591	−	0.807702i	$-0.299289\pi$
$263$	19.1231	1.17918	0.589591	+	0.807702i	$0.299289\pi$
$264$	0	0
$265$	4.00000	0.245718
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−13.3153	−0.811851	−0.405925	−	0.913906i	$-0.633051\pi$
$269$	−13.3153	−0.811851	−0.405925	+	0.913906i	$0.633051\pi$
$270$	0	0
$271$	24.0000	1.45790	0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000	1.45790	0.728948	+	0.684569i	$0.240010\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−3.12311	−0.188330
$276$	0	0
$277$	17.8078	1.06996	0.534982	−	0.844863i	$-0.320318\pi$
$277$	17.8078	1.06996	0.534982	+	0.844863i	$0.320318\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−5.12311	−0.305619	−0.152809	−	0.988256i	$-0.548832\pi$
$281$	−5.12311	−0.305619	−0.152809	+	0.988256i	$0.548832\pi$
$282$	0	0
$283$	3.12311	0.185649	0.0928247	−	0.995682i	$-0.470410\pi$
$283$	3.12311	0.185649	0.0928247	+	0.995682i	$0.470410\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	9.24621	0.543895
$290$	0	0
$291$	0	0
$292$	0	0
$293$	15.3693	0.897885	0.448943	−	0.893561i	$-0.351801\pi$
$293$	15.3693	0.897885	0.448943	+	0.893561i	$0.351801\pi$
$294$	0	0
$295$	20.4924	1.19311
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0.438447	0.0253561
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	8.49242	0.486275
$306$	0	0
$307$	−18.2462	−1.04137	−0.520683	−	0.853750i	$-0.674323\pi$
$307$	−18.2462	−1.04137	−0.520683	+	0.853750i	$0.674323\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−5.56155	−0.315367	−0.157683	−	0.987490i	$-0.550403\pi$
$311$	−5.56155	−0.315367	−0.157683	+	0.987490i	$0.550403\pi$
$312$	0	0
$313$	−15.3693	−0.868725	−0.434363	−	0.900738i	$-0.643026\pi$
$313$	−15.3693	−0.868725	−0.434363	+	0.900738i	$0.643026\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.75379	0.210834	0.105417	−	0.994428i	$-0.466382\pi$
$317$	3.75379	0.210834	0.105417	+	0.994428i	$0.466382\pi$
$318$	0	0
$319$	−11.1231	−0.622774
$320$	0	0
$321$	0	0
$322$	0	0
$323$	16.0000	0.890264
$324$	0	0
$325$	−0.438447	−0.0243207
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	31.4233	1.72718	0.863590	−	0.504194i	$-0.168211\pi$
$331$	31.4233	1.72718	0.863590	+	0.504194i	$0.168211\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−6.24621	−0.341267
$336$	0	0
$337$	−21.6155	−1.17747	−0.588736	−	0.808325i	$-0.700374\pi$
$337$	−21.6155	−1.17747	−0.588736	+	0.808325i	$0.700374\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−7.61553	−0.412404
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	16.4924	0.885360	0.442680	−	0.896680i	$-0.354028\pi$
$347$	16.4924	0.885360	0.442680	+	0.896680i	$0.354028\pi$
$348$	0	0
$349$	−34.6847	−1.85663	−0.928314	−	0.371798i	$-0.878741\pi$
$349$	−34.6847	−1.85663	−0.928314	+	0.371798i	$0.878741\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−15.5616	−0.828258	−0.414129	−	0.910218i	$-0.635914\pi$
$353$	−15.5616	−0.828258	−0.414129	+	0.910218i	$0.635914\pi$
$354$	0	0
$355$	27.1231	1.43954
$356$	0	0
$357$	0	0
$358$	0	0
$359$	23.6155	1.24638	0.623190	−	0.782071i	$-0.285836\pi$
$359$	23.6155	1.24638	0.623190	+	0.782071i	$0.285836\pi$
$360$	0	0
$361$	−9.24621	−0.486643
$362$	0	0
$363$	0	0
$364$	0	0
$365$	29.3693	1.53726
$366$	0	0
$367$	−14.2462	−0.743646	−0.371823	−	0.928304i	$-0.621267\pi$
$367$	−14.2462	−0.743646	−0.371823	+	0.928304i	$0.621267\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−10.4924	−0.543277	−0.271639	−	0.962399i	$-0.587566\pi$
$373$	−10.4924	−0.543277	−0.271639	+	0.962399i	$0.587566\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1.56155	−0.0804241
$378$	0	0
$379$	−12.4924	−0.641693	−0.320846	−	0.947131i	$-0.603967\pi$
$379$	−12.4924	−0.641693	−0.320846	+	0.947131i	$0.603967\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−9.75379	−0.498395	−0.249198	−	0.968453i	$-0.580167\pi$
$383$	−9.75379	−0.498395	−0.249198	+	0.968453i	$0.580167\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	23.3693	1.18487	0.592436	−	0.805618i	$-0.298166\pi$
$389$	23.3693	1.18487	0.592436	+	0.805618i	$0.298166\pi$
$390$	0	0
$391$	−5.12311	−0.259087
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−6.24621	−0.314281
$396$	0	0
$397$	16.4384	0.825022	0.412511	−	0.910953i	$-0.364652\pi$
$397$	16.4384	0.825022	0.412511	+	0.910953i	$0.364652\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	28.2462	1.41055	0.705274	−	0.708935i	$-0.250824\pi$
$401$	28.2462	1.41055	0.705274	+	0.708935i	$0.250824\pi$
$402$	0	0
$403$	−1.06913	−0.0532572
$404$	0	0
$405$	0	0
$406$	0	0
$407$	25.7538	1.27657
$408$	0	0
$409$	26.3002	1.30046	0.650230	−	0.759737i	$-0.274672\pi$
$409$	26.3002	1.30046	0.650230	+	0.759737i	$0.274672\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	28.4924	1.39864
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−20.4924	−1.00112	−0.500560	−	0.865702i	$-0.666873\pi$
$419$	−20.4924	−1.00112	−0.500560	+	0.865702i	$0.666873\pi$
$420$	0	0
$421$	−18.8769	−0.920004	−0.460002	−	0.887918i	$-0.652151\pi$
$421$	−18.8769	−0.920004	−0.460002	+	0.887918i	$0.652151\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	5.12311	0.248507
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−4.49242	−0.216392	−0.108196	−	0.994130i	$-0.534507\pi$
$431$	−4.49242	−0.216392	−0.108196	+	0.994130i	$0.534507\pi$
$432$	0	0
$433$	−24.7386	−1.18886	−0.594431	−	0.804146i	$-0.702623\pi$
$433$	−24.7386	−1.18886	−0.594431	+	0.804146i	$0.702623\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3.12311	−0.149398
$438$	0	0
$439$	−26.0540	−1.24349	−0.621744	−	0.783220i	$-0.713576\pi$
$439$	−26.0540	−1.24349	−0.621744	+	0.783220i	$0.713576\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	34.9309	1.65962	0.829808	−	0.558049i	$-0.188450\pi$
$443$	34.9309	1.65962	0.829808	+	0.558049i	$0.188450\pi$
$444$	0	0
$445$	22.7386	1.07791
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−8.24621	−0.389163	−0.194581	−	0.980886i	$-0.562335\pi$
$449$	−8.24621	−0.389163	−0.194581	+	0.980886i	$0.562335\pi$
$450$	0	0
$451$	30.6307	1.44234
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	19.3693	0.906058	0.453029	−	0.891496i	$-0.350343\pi$
$457$	19.3693	0.906058	0.453029	+	0.891496i	$0.350343\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−9.80776	−0.456793	−0.228397	−	0.973568i	$-0.573348\pi$
$461$	−9.80776	−0.456793	−0.228397	+	0.973568i	$0.573348\pi$
$462$	0	0
$463$	−20.4924	−0.952364	−0.476182	−	0.879347i	$-0.657980\pi$
$463$	−20.4924	−0.952364	−0.476182	+	0.879347i	$0.657980\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−9.36932	−0.433560	−0.216780	−	0.976220i	$-0.569555\pi$
$467$	−9.36932	−0.433560	−0.216780	+	0.976220i	$0.569555\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−24.9848	−1.14880
$474$	0	0
$475$	3.12311	0.143298
$476$	0	0
$477$	0	0
$478$	0	0
$479$	6.24621	0.285397	0.142698	−	0.989766i	$-0.454422\pi$
$479$	6.24621	0.285397	0.142698	+	0.989766i	$0.454422\pi$
$480$	0	0
$481$	3.61553	0.164854
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−22.7386	−1.03251
$486$	0	0
$487$	24.6847	1.11857	0.559284	−	0.828976i	$-0.311076\pi$
$487$	24.6847	1.11857	0.559284	+	0.828976i	$0.311076\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	7.80776	0.352359	0.176180	−	0.984358i	$-0.443626\pi$
$491$	7.80776	0.352359	0.176180	+	0.984358i	$0.443626\pi$
$492$	0	0
$493$	18.2462	0.821768
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	14.0540	0.629142	0.314571	−	0.949234i	$-0.398139\pi$
$499$	14.0540	0.629142	0.314571	+	0.949234i	$0.398139\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	17.3693	0.774460	0.387230	−	0.921983i	$-0.373432\pi$
$503$	17.3693	0.774460	0.387230	+	0.921983i	$0.373432\pi$
$504$	0	0
$505$	24.4924	1.08990
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−38.3002	−1.69763	−0.848813	−	0.528693i	$-0.822682\pi$
$509$	−38.3002	−1.69763	−0.848813	+	0.528693i	$0.822682\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	28.4924	1.25553
$516$	0	0
$517$	2.13826	0.0940406
$518$	0	0
$519$	0	0
$520$	0	0
$521$	21.6155	0.946993	0.473497	−	0.880796i	$-0.342992\pi$
$521$	21.6155	0.946993	0.473497	+	0.880796i	$0.342992\pi$
$522$	0	0
$523$	22.2462	0.972759	0.486379	−	0.873748i	$-0.338317\pi$
$523$	22.2462	0.972759	0.486379	+	0.873748i	$0.338317\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	12.4924	0.544178
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	4.30019	0.186262
$534$	0	0
$535$	22.2462	0.961788
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−21.8617	−0.941652
$540$	0	0
$541$	7.06913	0.303926	0.151963	−	0.988386i	$-0.451441\pi$
$541$	7.06913	0.303926	0.151963	+	0.988386i	$0.451441\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−26.2462	−1.12426
$546$	0	0
$547$	−26.5464	−1.13504	−0.567521	−	0.823359i	$-0.692097\pi$
$547$	−26.5464	−1.13504	−0.567521	+	0.823359i	$0.692097\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	11.1231	0.473860
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	9.12311	0.386558	0.193279	−	0.981144i	$-0.438088\pi$
$557$	9.12311	0.386558	0.193279	+	0.981144i	$0.438088\pi$
$558$	0	0
$559$	−3.50758	−0.148355
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−22.2462	−0.937566	−0.468783	−	0.883313i	$-0.655307\pi$
$563$	−22.2462	−0.937566	−0.468783	+	0.883313i	$0.655307\pi$
$564$	0	0
$565$	−5.75379	−0.242064
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−8.24621	−0.345699	−0.172850	−	0.984948i	$-0.555297\pi$
$569$	−8.24621	−0.345699	−0.172850	+	0.984948i	$0.555297\pi$
$570$	0	0
$571$	−44.4924	−1.86195	−0.930975	−	0.365083i	$-0.881041\pi$
$571$	−44.4924	−1.86195	−0.930975	+	0.365083i	$0.881041\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	−12.9309	−0.538319	−0.269160	−	0.963096i	$-0.586746\pi$
$577$	−12.9309	−0.538319	−0.269160	+	0.963096i	$0.586746\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−6.24621	−0.258692
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−14.0540	−0.580070	−0.290035	−	0.957016i	$-0.593667\pi$
$587$	−14.0540	−0.580070	−0.290035	+	0.957016i	$0.593667\pi$
$588$	0	0
$589$	7.61553	0.313792
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−4.73863	−0.194592	−0.0972962	−	0.995255i	$-0.531019\pi$
$593$	−4.73863	−0.194592	−0.0972962	+	0.995255i	$0.531019\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−8.00000	−0.326871	−0.163436	−	0.986554i	$-0.552258\pi$
$599$	−8.00000	−0.326871	−0.163436	+	0.986554i	$0.552258\pi$
$600$	0	0
$601$	−18.1922	−0.742077	−0.371038	−	0.928618i	$-0.620998\pi$
$601$	−18.1922	−0.742077	−0.371038	+	0.928618i	$0.620998\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2.49242	0.101331
$606$	0	0
$607$	34.7386	1.41000	0.704999	−	0.709208i	$-0.250948\pi$
$607$	34.7386	1.41000	0.704999	+	0.709208i	$0.250948\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0.300187	0.0121442
$612$	0	0
$613$	46.1080	1.86228	0.931141	−	0.364659i	$-0.118814\pi$
$613$	46.1080	1.86228	0.931141	+	0.364659i	$0.118814\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	30.9848	1.24740	0.623701	−	0.781663i	$-0.285628\pi$
$617$	30.9848	1.24740	0.623701	+	0.781663i	$0.285628\pi$
$618$	0	0
$619$	−4.49242	−0.180566	−0.0902829	−	0.995916i	$-0.528777\pi$
$619$	−4.49242	−0.180566	−0.0902829	+	0.995916i	$0.528777\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−42.2462	−1.68447
$630$	0	0
$631$	42.7386	1.70140	0.850699	−	0.525653i	$-0.176179\pi$
$631$	42.7386	1.70140	0.850699	+	0.525653i	$0.176179\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−1.36932	−0.0543397
$636$	0	0
$637$	−3.06913	−0.121603
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−11.3693	−0.449061	−0.224531	−	0.974467i	$-0.572085\pi$
$641$	−11.3693	−0.449061	−0.224531	+	0.974467i	$0.572085\pi$
$642$	0	0
$643$	1.36932	0.0540006	0.0270003	−	0.999635i	$-0.491404\pi$
$643$	1.36932	0.0540006	0.0270003	+	0.999635i	$0.491404\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−24.3002	−0.955339	−0.477669	−	0.878540i	$-0.658518\pi$
$647$	−24.3002	−0.955339	−0.477669	+	0.878540i	$0.658518\pi$
$648$	0	0
$649$	−32.0000	−1.25611
$650$	0	0
$651$	0	0
$652$	0	0
$653$	12.0540	0.471709	0.235854	−	0.971788i	$-0.424211\pi$
$653$	12.0540	0.471709	0.235854	+	0.971788i	$0.424211\pi$
$654$	0	0
$655$	6.63068	0.259082
$656$	0	0
$657$	0	0
$658$	0	0
$659$	14.2462	0.554954	0.277477	−	0.960732i	$-0.410502\pi$
$659$	14.2462	0.554954	0.277477	+	0.960732i	$0.410502\pi$
$660$	0	0
$661$	16.2462	0.631904	0.315952	−	0.948775i	$-0.397676\pi$
$661$	16.2462	0.631904	0.315952	+	0.948775i	$0.397676\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−3.56155	−0.137904
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−13.2614	−0.511949
$672$	0	0
$673$	−40.0540	−1.54397	−0.771984	−	0.635642i	$-0.780735\pi$
$673$	−40.0540	−1.54397	−0.771984	+	0.635642i	$0.780735\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−26.9848	−1.03711	−0.518556	−	0.855044i	$-0.673530\pi$
$677$	−26.9848	−1.03711	−0.518556	+	0.855044i	$0.673530\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−28.6847	−1.09759	−0.548794	−	0.835958i	$-0.684913\pi$
$683$	−28.6847	−1.09759	−0.548794	+	0.835958i	$0.684913\pi$
$684$	0	0
$685$	13.7538	0.525505
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−0.876894	−0.0334070
$690$	0	0
$691$	−44.9848	−1.71130	−0.855652	−	0.517551i	$-0.826844\pi$
$691$	−44.9848	−1.71130	−0.855652	+	0.517551i	$0.826844\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	6.63068	0.251516
$696$	0	0
$697$	−50.2462	−1.90321
$698$	0	0
$699$	0	0
$700$	0	0
$701$	27.8617	1.05232	0.526162	−	0.850385i	$-0.323631\pi$
$701$	27.8617	1.05232	0.526162	+	0.850385i	$0.323631\pi$
$702$	0	0
$703$	−25.7538	−0.971323
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−12.6307	−0.474355	−0.237178	−	0.971466i	$-0.576222\pi$
$709$	−12.6307	−0.474355	−0.237178	+	0.971466i	$0.576222\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−2.43845	−0.0913206
$714$	0	0
$715$	−2.73863	−0.102419
$716$	0	0
$717$	0	0
$718$	0	0
$719$	28.4924	1.06259	0.531294	−	0.847187i	$-0.321706\pi$
$719$	28.4924	1.06259	0.531294	+	0.847187i	$0.321706\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	3.56155	0.132273
$726$	0	0
$727$	−14.6307	−0.542622	−0.271311	−	0.962492i	$-0.587457\pi$
$727$	−14.6307	−0.542622	−0.271311	+	0.962492i	$0.587457\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	40.9848	1.51588
$732$	0	0
$733$	24.6307	0.909755	0.454878	−	0.890554i	$-0.349683\pi$
$733$	24.6307	0.909755	0.454878	+	0.890554i	$0.349683\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	9.75379	0.359285
$738$	0	0
$739$	12.6847	0.466613	0.233306	−	0.972403i	$-0.425046\pi$
$739$	12.6847	0.466613	0.233306	+	0.972403i	$0.425046\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−17.7538	−0.651323	−0.325662	−	0.945486i	$-0.605587\pi$
$743$	−17.7538	−0.651323	−0.325662	+	0.945486i	$0.605587\pi$
$744$	0	0
$745$	−40.4924	−1.48353
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−24.9848	−0.911710	−0.455855	−	0.890054i	$-0.650666\pi$
$751$	−24.9848	−0.911710	−0.455855	+	0.890054i	$0.650666\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	20.8769	0.759788
$756$	0	0
$757$	−34.4924	−1.25365	−0.626824	−	0.779161i	$-0.715646\pi$
$757$	−34.4924	−1.25365	−0.626824	+	0.779161i	$0.715646\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	30.6847	1.11232	0.556159	−	0.831076i	$-0.312275\pi$
$761$	30.6847	1.11232	0.556159	+	0.831076i	$0.312275\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−4.49242	−0.162212
$768$	0	0
$769$	−6.00000	−0.216366	−0.108183	−	0.994131i	$-0.534503\pi$
$769$	−6.00000	−0.216366	−0.108183	+	0.994131i	$0.534503\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−27.3693	−0.984406	−0.492203	−	0.870480i	$-0.663808\pi$
$773$	−27.3693	−0.984406	−0.492203	+	0.870480i	$0.663808\pi$
$774$	0	0
$775$	2.43845	0.0875916
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−30.6307	−1.09746
$780$	0	0
$781$	−42.3542	−1.51555
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−42.2462	−1.50783
$786$	0	0
$787$	−9.75379	−0.347685	−0.173843	−	0.984773i	$-0.555618\pi$
$787$	−9.75379	−0.347685	−0.173843	+	0.984773i	$0.555618\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−1.86174	−0.0661123
$794$	0	0
$795$	0	0
$796$	0	0
$797$	4.24621	0.150409	0.0752043	−	0.997168i	$-0.476039\pi$
$797$	4.24621	0.150409	0.0752043	+	0.997168i	$0.476039\pi$
$798$	0	0
$799$	−3.50758	−0.124089
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−45.8617	−1.61843
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−26.0000	−0.914111	−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000	−0.914111	−0.457056	+	0.889438i	$0.651096\pi$
$810$	0	0
$811$	5.06913	0.178001	0.0890006	−	0.996032i	$-0.471633\pi$
$811$	5.06913	0.178001	0.0890006	+	0.996032i	$0.471633\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−37.8617	−1.32624
$816$	0	0
$817$	24.9848	0.874109
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−24.7386	−0.863384	−0.431692	−	0.902021i	$-0.642083\pi$
$821$	−24.7386	−0.863384	−0.431692	+	0.902021i	$0.642083\pi$
$822$	0	0
$823$	53.5616	1.86704	0.933519	−	0.358527i	$-0.116721\pi$
$823$	53.5616	1.86704	0.933519	+	0.358527i	$0.116721\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	28.4924	0.990779	0.495389	−	0.868671i	$-0.335025\pi$
$827$	28.4924	0.990779	0.495389	+	0.868671i	$0.335025\pi$
$828$	0	0
$829$	−3.75379	−0.130374	−0.0651872	−	0.997873i	$-0.520764\pi$
$829$	−3.75379	−0.130374	−0.0651872	+	0.997873i	$0.520764\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	35.8617	1.24254
$834$	0	0
$835$	−32.0000	−1.10741
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−33.7538	−1.16531	−0.582655	−	0.812720i	$-0.697986\pi$
$839$	−33.7538	−1.16531	−0.582655	+	0.812720i	$0.697986\pi$
$840$	0	0
$841$	−16.3153	−0.562598
$842$	0	0
$843$	0	0
$844$	0	0
$845$	25.6155	0.881201
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	8.24621	0.282676
$852$	0	0
$853$	30.9848	1.06090	0.530450	−	0.847716i	$-0.322023\pi$
$853$	30.9848	1.06090	0.530450	+	0.847716i	$0.322023\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−12.4384	−0.424889	−0.212445	−	0.977173i	$-0.568143\pi$
$857$	−12.4384	−0.424889	−0.212445	+	0.977173i	$0.568143\pi$
$858$	0	0
$859$	54.0540	1.84430	0.922149	−	0.386835i	$-0.126432\pi$
$859$	54.0540	1.84430	0.922149	+	0.386835i	$0.126432\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−37.1771	−1.26552	−0.632761	−	0.774347i	$-0.718079\pi$
$863$	−37.1771	−1.26552	−0.632761	+	0.774347i	$0.718079\pi$
$864$	0	0
$865$	−20.0000	−0.680020
$866$	0	0
$867$	0	0
$868$	0	0
$869$	9.75379	0.330875
$870$	0	0
$871$	1.36932	0.0463975
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−34.9848	−1.18135	−0.590677	−	0.806908i	$-0.701139\pi$
$877$	−34.9848	−1.18135	−0.590677	+	0.806908i	$0.701139\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−13.1231	−0.442129	−0.221064	−	0.975259i	$-0.570953\pi$
$881$	−13.1231	−0.442129	−0.221064	+	0.975259i	$0.570953\pi$
$882$	0	0
$883$	−28.9848	−0.975418	−0.487709	−	0.873006i	$-0.662167\pi$
$883$	−28.9848	−0.975418	−0.487709	+	0.873006i	$0.662167\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−7.31534	−0.245625	−0.122813	−	0.992430i	$-0.539191\pi$
$887$	−7.31534	−0.245625	−0.122813	+	0.992430i	$0.539191\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−2.13826	−0.0715542
$894$	0	0
$895$	−6.63068	−0.221639
$896$	0	0
$897$	0	0
$898$	0	0
$899$	8.68466	0.289650
$900$	0	0
$901$	10.2462	0.341351
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−7.50758	−0.249560
$906$	0	0
$907$	51.1231	1.69751	0.848757	−	0.528782i	$-0.177351\pi$
$907$	51.1231	1.69751	0.848757	+	0.528782i	$0.177351\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	28.8769	0.956734	0.478367	−	0.878160i	$-0.341229\pi$
$911$	28.8769	0.956734	0.478367	+	0.878160i	$0.341229\pi$
$912$	0	0
$913$	−44.4924	−1.47248
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	54.2462	1.78942	0.894709	−	0.446650i	$-0.147383\pi$
$919$	54.2462	1.78942	0.894709	+	0.446650i	$0.147383\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−5.94602	−0.195716
$924$	0	0
$925$	−8.24621	−0.271134
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−14.1922	−0.465632	−0.232816	−	0.972521i	$-0.574794\pi$
$929$	−14.1922	−0.465632	−0.232816	+	0.972521i	$0.574794\pi$
$930$	0	0
$931$	21.8617	0.716490
$932$	0	0
$933$	0	0
$934$	0	0
$935$	32.0000	1.04651
$936$	0	0
$937$	32.2462	1.05344	0.526719	−	0.850040i	$-0.323422\pi$
$937$	32.2462	1.05344	0.526719	+	0.850040i	$0.323422\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−30.8769	−1.00656	−0.503279	−	0.864124i	$-0.667873\pi$
$941$	−30.8769	−1.00656	−0.503279	+	0.864124i	$0.667873\pi$
$942$	0	0
$943$	9.80776	0.319385
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.56155	−0.0507436	−0.0253718	−	0.999678i	$-0.508077\pi$
$947$	−1.56155	−0.0507436	−0.0253718	+	0.999678i	$0.508077\pi$
$948$	0	0
$949$	−6.43845	−0.209001
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−27.7538	−0.899033	−0.449517	−	0.893272i	$-0.648404\pi$
$953$	−27.7538	−0.899033	−0.449517	+	0.893272i	$0.648404\pi$
$954$	0	0
$955$	50.7386	1.64186
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−25.0540	−0.808193
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0.876894	0.0282282
$966$	0	0
$967$	−2.05398	−0.0660514	−0.0330257	−	0.999455i	$-0.510514\pi$
$967$	−2.05398	−0.0660514	−0.0330257	+	0.999455i	$0.510514\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−6.63068	−0.212789	−0.106394	−	0.994324i	$-0.533931\pi$
$971$	−6.63068	−0.212789	−0.106394	+	0.994324i	$0.533931\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−25.2311	−0.807213	−0.403607	−	0.914933i	$-0.632244\pi$
$977$	−25.2311	−0.807213	−0.403607	+	0.914933i	$0.632244\pi$
$978$	0	0
$979$	−35.5076	−1.13483
$980$	0	0
$981$	0	0
$982$	0	0
$983$	52.1080	1.66199	0.830993	−	0.556283i	$-0.187773\pi$
$983$	52.1080	1.66199	0.830993	+	0.556283i	$0.187773\pi$
$984$	0	0
$985$	−33.8617	−1.07892
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−8.00000	−0.254385
$990$	0	0
$991$	−56.9848	−1.81018	−0.905092	−	0.425217i	$-0.860198\pi$
$991$	−56.9848	−1.81018	−0.905092	+	0.425217i	$0.860198\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−22.2462	−0.705252
$996$	0	0
$997$	14.0000	0.443384	0.221692	−	0.975117i	$-0.428842\pi$
$997$	14.0000	0.443384	0.221692	+	0.975117i	$0.428842\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1656.2.a.j.1.2		2
3.2	odd	2		184.2.a.e.1.2	✓	2
4.3	odd	2		3312.2.a.t.1.1		2
12.11	even	2		368.2.a.i.1.1		2
15.2	even	4		4600.2.e.o.4049.2		4
15.8	even	4		4600.2.e.o.4049.3		4
15.14	odd	2		4600.2.a.s.1.1		2
21.20	even	2		9016.2.a.w.1.1		2
24.5	odd	2		1472.2.a.u.1.1		2
24.11	even	2		1472.2.a.p.1.2		2
60.59	even	2		9200.2.a.br.1.2		2
69.68	even	2		4232.2.a.o.1.2		2
276.275	odd	2		8464.2.a.bd.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
184.2.a.e.1.2	✓	2	3.2	odd	2
368.2.a.i.1.1		2	12.11	even	2
1472.2.a.p.1.2		2	24.11	even	2
1472.2.a.u.1.1		2	24.5	odd	2
1656.2.a.j.1.2		2	1.1	even	1	trivial
3312.2.a.t.1.1		2	4.3	odd	2
4232.2.a.o.1.2		2	69.68	even	2
4600.2.a.s.1.1		2	15.14	odd	2
4600.2.e.o.4049.2		4	15.2	even	4
4600.2.e.o.4049.3		4	15.8	even	4
8464.2.a.bd.1.1		2	276.275	odd	2
9016.2.a.w.1.1		2	21.20	even	2
9200.2.a.br.1.2		2	60.59	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1656 = 2^{3} \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1656.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{5} +O(q^{10})\)	\(q-2.00000 q^{5} +3.12311 q^{11} +0.438447 q^{13} -5.12311 q^{17} -3.12311 q^{19} +1.00000 q^{23} -1.00000 q^{25} -3.56155 q^{29} -2.43845 q^{31} +8.24621 q^{37} +9.80776 q^{41} -8.00000 q^{43} +0.684658 q^{47} -7.00000 q^{49} -2.00000 q^{53} -6.24621 q^{55} -10.2462 q^{59} -4.24621 q^{61} -0.876894 q^{65} +3.12311 q^{67} -13.5616 q^{71} -14.6847 q^{73} +3.12311 q^{79} -14.2462 q^{83} +10.2462 q^{85} -11.3693 q^{89} +6.24621 q^{95} +11.3693 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5}+O(q^{10}) \) 2 * q - 4 * q^5	\( 2 q - 4 q^{5} - 2 q^{11} + 5 q^{13} - 2 q^{17} + 2 q^{19} + 2 q^{23} - 2 q^{25} - 3 q^{29} - 9 q^{31} - q^{41} - 16 q^{43} - 11 q^{47} - 14 q^{49} - 4 q^{53} + 4 q^{55} - 4 q^{59} + 8 q^{61} - 10 q^{65} - 2 q^{67} - 23 q^{71} - 17 q^{73} - 2 q^{79} - 12 q^{83} + 4 q^{85} + 2 q^{89} - 4 q^{95} - 2 q^{97}+O(q^{100}) \) 2 * q - 4 * q^5 - 2 * q^11 + 5 * q^13 - 2 * q^17 + 2 * q^19 + 2 * q^23 - 2 * q^25 - 3 * q^29 - 9 * q^31 - q^41 - 16 * q^43 - 11 * q^47 - 14 * q^49 - 4 * q^53 + 4 * q^55 - 4 * q^59 + 8 * q^61 - 10 * q^65 - 2 * q^67 - 23 * q^71 - 17 * q^73 - 2 * q^79 - 12 * q^83 + 4 * q^85 + 2 * q^89 - 4 * q^95 - 2 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more